To find the area of the region bounded by the parabola $y = 2x^2$, the tangent line to this parabola at $(2,8)$, and the x-axis, we can follow these steps:

Step 1: Find the Equation of the Tangent Line

1. Differentiate $y = 2x^2$ with respect to $x$ to get the slope of the tangent. $\frac{dy}{dx} = 4x$
2. At $x = 2$, the slope of the tangent is: $4 \cdot 2 = 8$
3. The equation of the tangent line at $(2,8)$ is: $y - 8 = 8(x - 2)$ Simplifying this, we get: $y = 8x - 8$

Step 2: Set up the Integrals

Now, we need to find the points where the tangent line and the parabola intersect with the x-axis:

1. For the tangent line $y = 8x - 8$, set $y = 0$ to find the x-intercept: $0 = 8x - 8 \Rightarrow x = 1$ So the tangent line intersects the x-axis at $(1,0)$.

2. For the parabola $y = 2x^2$, set $y = 0$ to find the x-intercepts: $0 = 2x^2 \Rightarrow x = 0$ So the parabola intersects the x-axis at $(0,0)$.

The region bounded by the parabola, the tangent, and the x-axis lies between $x = 0$ and $x = 1$.

Step 3: Integrate to Find the Area

To find the area between the parabola and the tangent line from $x = 0$ to $x = 1$, we set up the integral: $\text{Area} = \int_0^1 \left((8x - 8) - (2x^2)\right) \, dx$

Step 4: Simplify the Integral

1. Simplify the expression inside the integral: $\int_0^1 (8x - 8 - 2x^2) \, dx = \int_0^1 (-2x^2 + 8x - 8) \, dx$
2. Integrate term by term: $= \left[ -\frac{2}{3}x^3 + 4x^2 - 8x \right]_0^1$
3. Substitute the limits: $= \left( -\frac{2}{3}(1)^3 + 4(1)^2 - 8(1) \right) - \left( -\frac{2}{3}(0)^3 + 4(0)^2 - 8(0) \right)$ $= -\frac{2}{3} + 4 - 8$ $= -\frac{2}{3} - 4 = -\frac{14}{3}$

Since area cannot be negative, take the absolute value: $\text{Area} = \frac{14}{3}$

Final Answer

The area of the region bounded by the parabola $y = 2x^2$, the tangent line at $(2,8)$, and the x-axis is: $\frac{14}{3} \text{ square units}$

Would you like more details on any specific step, or have any other questions?

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1. How do you find the tangent line to a curve at a specific point?
2. What other methods could be used to find the area between curves?
3. How do you interpret the result of an integral in the context of areas?
4. What is the significance of taking the absolute value when computing areas?
5. How can this problem be visualized geometrically?

Tip: For bounded areas between curves, always ensure you correctly identify intersections and orientation of the area to avoid sign errors.